Problem 24 - College Algebra with Modeling and Visualization

Question

The sum of an arithmetic series with 20 terms is \(610 .\) If \(a_{20}=59,\) find \(a_{1}.\)

Step-by-Step Solution

Answer

The first term of the arithmetic series, \( a_1 \), is 2.

Step 1: Recall the Formula for the Sum of an Arithmetic Series

The sum \( S_n \) of an arithmetic series with \( n \) terms is given by the formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \] where \( a_1 \) is the first term and \( a_n \) is the last term.

Step 2: Substitute Known Values into the Sum Formula

We know from the problem statement that \( S_{20} = 610 \), \( n = 20 \), and \( a_{20} = 59 \). Substitute these values into the sum formula: \[ 610 = \frac{20}{2} (a_1 + 59) \]

Step 3: Simplify the Equation

Simplify \( \frac{20}{2} \) to \( 10 \), leading to: \[ 610 = 10(a_1 + 59) \]

Step 4: Isolate \( a_1 \)

Divide both sides by 10 to isolate the expression containing \( a_1 \): \[ a_1 + 59 = \frac{610}{10} = 61 \]. Then, isolate \( a_1 \) by subtracting 59 from both sides: \[ a_1 = 61 - 59 \]

Step 5: Calculate \( a_1 \)

Perform the subtraction: \( a_1 = 2 \). Thus, the first term of the arithmetic series is 2.

Key Concepts

Sum of Arithmetic Series

The sum of an arithmetic series is the total when all the terms of the series are added. This is an important concept as it allows us to combine the elements of a sequence to find their total contribution. The formula for calculating the sum of an arithmetic series with \( n \) terms is:

• \[ S_n = \frac{n}{2} (a_1 + a_n) \]

Here, \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the last term. This formula is helpful because it simplifies adding up all the terms individually.

Using this formula, we can find the sum if we know the first and last terms and the total number of terms. For example, in our exercise, we knew that the sum of the first 20 terms was 610, which allowed us to figure out other missing parts of the sequence.

First Term of Arithmetic Series

In any arithmetic series, the first term \( a_1 \) is crucial in defining the sequence. It sets the starting point for the series. From this term, each subsequent term is obtained by adding a constant difference.

To find \( a_1 \) when you have other information like the sum of the series, you can rearrange the sum formula. For instance, in our given problem, we used:

• Given: \( S_{20} = 610 \), \( a_{20} = 59 \)
• Substitute into sum formula: \[ 610 = \frac{20}{2} (a_1 + 59) \]
• Simplify: \[ 610 = 10(a_1 + 59) \]
• Isolate \( a_1 \) by solving: \[ a_1 = 2 \]

This shows how the first term can often be deduced when combined with the series sum and other terms.

Arithmetic Sequence Calculation

Arithmetic sequences have a specific pattern where each term is obtained by adding a constant amount, known as the common difference, to the previous term. This makes calculations predictable.

To calculate terms in the arithmetic sequence, we use the formula:

• General formula: \[ a_n = a_1 + (n - 1) \cdot d \]
• \( a_n \): the nth term
• \( a_1 \): the first term
• \( d \): common difference

In our exercise, while we were mainly dealing with the sum and first term, knowing \( a_{20} = 59 \) and \( n = 20 \), it'd be possible to find the common difference if needed by rearranging the sequence formula or confirming the calculations through this approach.

Understanding such formulae allows for solving many different aspects of arithmetic sequences effectively, gaining insight into both individual terms and the sequence as a whole.